The Regularity of the Boundary of a Multidimensional Aggregation Patch

Let d ≥ 2 and let N(y) be the fundamental solution of the Laplace equation in R. We consider the aggregation equation ∂ρ ∂t + div(ρv) = 0, v = −∇N ∗ ρ with initial data ρ(x, 0) = χD0 where χD0 is the indicator function of a bounded domain D0 ⊂ R. We fix 0 < γ < 1 and take D0 to be a bounded C 1+γ domain (i. e. a domain with smooth boundary of class C). In this paper we prove the following theorem: If D0 is a C 1+γ domain, then the initial value problem above has a solution given by ρ(x, t) = 1 1−t χ Dt(x), x ∈ R, 0 ≤ t < 1 where Dt is a C domain for all 0 ≤ t < 1.